MARSDIAL: Show Me the Way to Go Home
 
Students will develop skills in working with geometric measuring tools in a realworld application by measuring shadow lengths and angles and by using this information to achieve the goal of finding direction.
REALWORLD APPLICATION:
Students will learn how to find true north so that they can begin to navigate without the use of modern technology.
BACKGROUND:
In this activity, students will learn how to determine the direction of true north using only a shadow cast by the Sun. If the students have performed Marsdial Activity #1, they will already be well acquainted with the materials they will be using in this activity.
PROCEDURE:
The general procedure is very simple. Beginning about one to one and half hours before local solar noon (which will occur between 11:30 AM and 12:30 PM, the exact time isn't important), place the pencil perpendicular to the ground in a marked spot (either on a piece of paper or using sidewalk chalk) and trace its shadow. Accuracy is very important. Measure the length of the shadow in either English or S.I. (metric) units. As time passes, the shadow length will change. Measure the shadow periodically over the next three hours until the shadow length is again exactly equal to the initial measurement. Trace the shadow as before.The line which bisects the angle formed by the two traced shadows is exactly true north! The simplest way to bisect the angle is to simply measure the angle with a protractor and draw a line at exactly half that angle. The method works because the shadow length is always symmetrical about local solar noon (Geometry teachers may want to discuss periodic functions in conjunction with this activity). You should start measuring at least an hour before solar noon because as solar noon approaches, the changes in the shadow length become much smaller and therefore require much more accuracy in measurement. There are a number of extensions to this activity that could be made as well; see the Extensions section below.
ASSESSMENT:
Using a known reference (a magnetic compass will not show true north), figure out in advance where true north is. A GPS system will show you, or you can simply do this activity yourself ahead of time  it's extremely accurate! Compare how close your students came to finding true north.ADAPTATIONS / EXTENSIONS:
 Geometry students can use the compass procedure to bisect the angle by construction. Using this method and a stick which has been cut to be exactly the initial shadow length, students can find true north in exactly the same way the ancients did!
 The time at which the shadow points to true north is at local solar noon. By recording the clock time at which this occurs, the students can figure out how many degrees of longitude they are from the "reference longitude" that establishes there time zone. Four minutes of difference represents one degree of longitude away from the reference longitude.
 The approximate direction of true north can be found any time by rotating a sundial until the shadow on the dial points to the current time. The 12:00 marking will then point to true north. The problem with this is that local solar time and clock time are not the same, so you are introducing a good bit of error. Have your students do both methods and compare their results. This method is much faster, but is less accurate. Is the tradeoff worth it?
National Council of Mathematics Teachers Principles and Standards:
Geometry: Use Visualization to Solve Problems 

Measurement: Understand Measurable Attributes 

Measure: Apply Appropriate Tools, Techniques, and Formulae 
